Operating Characteristics of Nozzles P M V Subbarao Professor Mechanical Engineering Department I I T Delhi From Takeoff to cruising …… Realizing New Events of Physics…….

Converging Nozzle p0p0 pbpb p b = Back Pressure Design Variables: Outlet Condition:

Designed Exit Conditions Under design conditions the pressure at the exit plane of the nozzle is applied back pressure.

Profile of the Nozzle At design Conditions:

Full Capacity Convergent Nozzle

Remarks on Isentropic Nozzle Design Length of the nozzle is immaterial for an isentropic nozzle. Strength requirements of nozzle material may decide the nozzle length. Either Mach number variation or Area variation or Pressure variation is specified as a function or arbitrary length unit. Nozzle design attains maximum capacity when the exit Mach number is unity.

Converging Nozzle p0p0 P b,critical

Operational Characteristics of Nozzles A variable area passage designed to accelerate the a gas flow is considered for study. The concern here is with the effect of changes in the upstream and downstream pressures on the nature of the inside flow and on the mass flow rate through a nozzle. Four different cases considered for analysis are: Converging nozzle with constant upstream conditions. Converging-diverging nozzle with constant upstream conditions. Converging nozzle with constant downstream conditions. Converging-diverging nozzle with constant downstream conditions.

Pressure Distribution in Under Expanded Nozzle p0p0 P b,critical p b =p 0 p b,critical< p b1< p 0 p b,critical< p b2< p 0 p b,critical< p b3< p 0 At all the above conditions, the pressure at the exit plane of nozzle, p exit = p b.

Variation of Mass Flow Rate in Exit Pressure 1 1

Variation of in Exit Pressure 1 1

Variation of in Mass Flow Rate 1

Low Back Pressure Operation

Convergent-Divergent Nozzle Under Design Conditions

Convergent-Divergent Nozzle with High Back Pressure p * < p b1< p 0 p throat> p *

Convergent-Divergent Nozzle with High Back Pressure When p b is very nearly the same as p 0 the flow remains subsonic throughout. The flow in the nozzle is then similar to that in a venturi. The local pressure drops from p 0 to a minimum value at the throat, p throat, which is greater than p *. The local pressure increases from throat to exit plane of the nozzle. The pressure at the exit plate of the nozzle is equal to the back pressure. This trend will continue for a particular value of back pressure.

Convergent-Divergent Nozzle with High Back Pressure At all these back pressures the exit plane pressure is equal to the back pressure. p throat> p *

At exit with high back pressure p b At throat with high back pressure p b

For a given value of high back pressure corresponding throat pressure can be calculated. As exit area is higher than throat area throat pressure is always less than exit plane pressure. An decreasing exit pressure produces lowering throat pressure

Variation of Mass Flow Rate in Exit Pressure 1 1

Variation of in Mass Flow Rate 1

Numerical Solution for Mach Number Caluculation Use “Newton’s Method” to extract numerical solution At correct Mach number (for given A/A*) … Define: Expand F(M) is Taylor’s series about some arbitrary Mach number M (j)

Solve for M

From Earlier Definition, thus if M (j) is chosen to be “close” to M And we can truncate after the first order terms with “little” Loss of accuracy Still exact expression

First Order approximation of solution for M However; one would anticipate that “Hat” indicates that solution is no longer exact “estimate is closer than original guess”

And we would anticipate that “refined estimate” …. Iteration 1 If we substitute back into the approximate expression

Abstracting to a “j th ” iteration Iterate until convergence j={0,1,….} Drop from loop when